MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem6 Unicode version

Theorem fin1a2lem6 8241
Description: Lemma for fin1a2 8251. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
fin1a2lem.aa  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem6  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )

Proof of Theorem fin1a2lem6
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
21fin1a2lem2 8237 . . 3  |-  S : On
-1-1-> On
3 fin1a2lem.b . . . . 5  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
43fin1a2lem4 8239 . . . 4  |-  E : om
-1-1-> om
5 f1f 5598 . . . 4  |-  ( E : om -1-1-> om  ->  E : om --> om )
6 frn 5556 . . . . 5  |-  ( E : om --> om  ->  ran 
E  C_  om )
7 omsson 4808 . . . . 5  |-  om  C_  On
86, 7syl6ss 3320 . . . 4  |-  ( E : om --> om  ->  ran 
E  C_  On )
94, 5, 8mp2b 10 . . 3  |-  ran  E  C_  On
10 f1ores 5648 . . 3  |-  ( ( S : On -1-1-> On  /\ 
ran  E  C_  On )  ->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
) )
112, 9, 10mp2an 654 . 2  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E )
129sseli 3304 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
b  e.  On )
131fin1a2lem1 8236 . . . . . . . . 9  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
1412, 13syl 16 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( S `  b
)  =  suc  b
)
1514eqeq1d 2412 . . . . . . 7  |-  ( b  e.  ran  E  -> 
( ( S `  b )  =  a  <->  suc  b  =  a
) )
1615rexbiia 2699 . . . . . 6  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  E. b  e.  ran  E  suc  b  =  a )
174, 5, 6mp2b 10 . . . . . . . . . . . 12  |-  ran  E  C_ 
om
1817sseli 3304 . . . . . . . . . . 11  |-  ( b  e.  ran  E  -> 
b  e.  om )
19 peano2 4824 . . . . . . . . . . 11  |-  ( b  e.  om  ->  suc  b  e.  om )
2018, 19syl 16 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  suc  b  e.  om )
213fin1a2lem5 8240 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  e.  ran  E  <->  -. 
suc  b  e.  ran  E ) )
2221biimpd 199 . . . . . . . . . . 11  |-  ( b  e.  om  ->  (
b  e.  ran  E  ->  -.  suc  b  e. 
ran  E ) )
2318, 22mpcom 34 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  -.  suc  b  e.  ran  E )
2420, 23jca 519 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
( suc  b  e.  om 
/\  -.  suc  b  e. 
ran  E ) )
25 eleq1 2464 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( suc  b  e.  om  <->  a  e.  om ) )
26 eleq1 2464 . . . . . . . . . . 11  |-  ( suc  b  =  a  -> 
( suc  b  e.  ran  E  <->  a  e.  ran  E ) )
2726notbid 286 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( -.  suc  b  e.  ran  E  <->  -.  a  e.  ran  E ) )
2825, 27anbi12d 692 . . . . . . . . 9  |-  ( suc  b  =  a  -> 
( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  <-> 
( a  e.  om  /\ 
-.  a  e.  ran  E ) ) )
2924, 28syl5ibcom 212 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( suc  b  =  a  ->  ( a  e. 
om  /\  -.  a  e.  ran  E ) ) )
3029rexlimiv 2784 . . . . . . 7  |-  ( E. b  e.  ran  E  suc  b  =  a  ->  ( a  e.  om  /\ 
-.  a  e.  ran  E ) )
31 peano1 4823 . . . . . . . . . . . . . 14  |-  (/)  e.  om
323fin1a2lem3 8238 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  ( E `  (/) )  =  ( 2o  .o  (/) ) )
3331, 32ax-mp 8 . . . . . . . . . . . . 13  |-  ( E `
 (/) )  =  ( 2o  .o  (/) )
34 om0x 6722 . . . . . . . . . . . . 13  |-  ( 2o 
.o  (/) )  =  (/)
3533, 34eqtri 2424 . . . . . . . . . . . 12  |-  ( E `
 (/) )  =  (/)
36 f1fun 5600 . . . . . . . . . . . . . 14  |-  ( E : om -1-1-> om  ->  Fun 
E )
374, 36ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  E
38 f1dm 5602 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  dom 
E  =  om )
394, 38ax-mp 8 . . . . . . . . . . . . . 14  |-  dom  E  =  om
4031, 39eleqtrri 2477 . . . . . . . . . . . . 13  |-  (/)  e.  dom  E
41 fvelrn 5825 . . . . . . . . . . . . 13  |-  ( ( Fun  E  /\  (/)  e.  dom  E )  ->  ( E `  (/) )  e.  ran  E )
4237, 40, 41mp2an 654 . . . . . . . . . . . 12  |-  ( E `
 (/) )  e.  ran  E
4335, 42eqeltrri 2475 . . . . . . . . . . 11  |-  (/)  e.  ran  E
44 eleq1 2464 . . . . . . . . . . 11  |-  ( a  =  (/)  ->  ( a  e.  ran  E  <->  (/)  e.  ran  E ) )
4543, 44mpbiri 225 . . . . . . . . . 10  |-  ( a  =  (/)  ->  a  e. 
ran  E )
4645necon3bi 2608 . . . . . . . . 9  |-  ( -.  a  e.  ran  E  ->  a  =/=  (/) )
47 nnsuc 4821 . . . . . . . . 9  |-  ( ( a  e.  om  /\  a  =/=  (/) )  ->  E. b  e.  om  a  =  suc  b )
4846, 47sylan2 461 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  om  a  =  suc  b )
49 eleq1 2464 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( a  e.  om  <->  suc  b  e.  om )
)
50 eleq1 2464 . . . . . . . . . . . . . . . . 17  |-  ( a  =  suc  b  -> 
( a  e.  ran  E  <->  suc  b  e.  ran  E ) )
5150notbid 286 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( -.  a  e. 
ran  E  <->  -.  suc  b  e. 
ran  E ) )
5249, 51anbi12d 692 . . . . . . . . . . . . . . 15  |-  ( a  =  suc  b  -> 
( ( a  e. 
om  /\  -.  a  e.  ran  E )  <->  ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E ) ) )
5352anbi1d 686 . . . . . . . . . . . . . 14  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om ) 
<->  ( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om ) ) )
54 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  -.  suc  b  e. 
ran  E )
5521adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  ( b  e.  ran  E  <->  -.  suc  b  e.  ran  E ) )
5654, 55mpbird 224 . . . . . . . . . . . . . 14  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E
)
5753, 56syl6bi 220 . . . . . . . . . . . . 13  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E ) )
5857com12 29 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  b  e. 
om )  ->  (
a  =  suc  b  ->  b  e.  ran  E
) )
5958impr 603 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  b  e.  ran  E )
60 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  a  =  suc  b )
6160eqcomd 2409 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  suc  b  =  a )
6259, 61jca 519 . . . . . . . . . 10  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) )
6362ex 424 . . . . . . . . 9  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( (
b  e.  om  /\  a  =  suc  b )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) ) )
6463reximdv2 2775 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( E. b  e.  om  a  =  suc  b  ->  E. b  e.  ran  E  suc  b  =  a ) )
6548, 64mpd 15 . . . . . . 7  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  ran  E  suc  b  =  a )
6630, 65impbii 181 . . . . . 6  |-  ( E. b  e.  ran  E  suc  b  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
6716, 66bitri 241 . . . . 5  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
68 f1fn 5599 . . . . . . 7  |-  ( S : On -1-1-> On  ->  S  Fn  On )
692, 68ax-mp 8 . . . . . 6  |-  S  Fn  On
70 fvelimab 5741 . . . . . 6  |-  ( ( S  Fn  On  /\  ran  E  C_  On )  ->  ( a  e.  ( S " ran  E
)  <->  E. b  e.  ran  E ( S `  b
)  =  a ) )
7169, 9, 70mp2an 654 . . . . 5  |-  ( a  e.  ( S " ran  E )  <->  E. b  e.  ran  E ( S `
 b )  =  a )
72 eldif 3290 . . . . 5  |-  ( a  e.  ( om  \  ran  E )  <->  ( a  e. 
om  /\  -.  a  e.  ran  E ) )
7367, 71, 723bitr4i 269 . . . 4  |-  ( a  e.  ( S " ran  E )  <->  a  e.  ( om  \  ran  E
) )
7473eqriv 2401 . . 3  |-  ( S
" ran  E )  =  ( om  \  ran  E )
75 f1oeq3 5626 . . 3  |-  ( ( S " ran  E
)  =  ( om 
\  ran  E )  ->  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
)  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E
) ) )
7674, 75ax-mp 8 . 2  |-  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S
" ran  E )  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om 
\  ran  E )
)
7711, 76mpbi 200 1  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    \ cdif 3277    C_ wss 3280   (/)c0 3588    e. cmpt 4226   Oncon0 4541   suc csuc 4543   omcom 4804   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   2oc2o 6677    .o comu 6681
This theorem is referenced by:  fin1a2lem7  8242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688
  Copyright terms: Public domain W3C validator